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See also: conditional probability 

Bayesian Rule

The Bayesian rule, as developed below, enables us to compute the probability of an event by first determining whether or not some other event has occurred. In some situations, it is easier to compute the probability of an event as soon as we know if some other event has occurred.

Given the events A and B, we can express A = (A B) (A B'):
 

A B
 A not B =
( A B ) ( A B' ) =
A B
A not B
 =
baye's rule
A

Since (A B) and (A B') are mutually exclusive, their probabilities can simply be added to obtain the probability of P(A):

P(A) = P (A B) + P(A B') = P(A|B) . P(B) + P(A|B') . P(B')
P(A) = P(A|B) . P(B) + P(A|B') . [1-P(B)]

The probability of the event A is the weighted average of the conditional probabilities of A given B, and A given not B. The weights for the conditional probabilities are defined by the probabilities of the conditional events B and not B.
 

Bayes' Formula:

The rule, developed above for the two events A and B, can also be generalized for more than two events. In case the sample space consists of n mutually exclusive events Fi, the probability for event E is:

P(E) is the weighted average of P(E|Fi), the weight being the probability of the event Fi on which it is conditioned.

When we interpret the Fj as hypotheses about a question, Bayes' formula shows us how the evidence should change our opinion held prior to the experiments.

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